Issolvi ∫ x^2(20x^7-7x^4+1) wrt x

$$\int x ^ { 2 } ( 20 x ^ { 7 } - 7 x ^ { 4 } + 1 ) d x$$

Evalwa

$$2x^{10}-x^{7}+\frac{x^{3}}{3}+С$$

Passi tas-Soluzzjonijiet

Uża l-propjetà distributtiva biex timmultiplika x^{2} b'20x^{7}-7x^{4}+1.

$$\int 20x^{9}-7x^{6}+x^{2}\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int 20x^{9}\mathrm{d}x+\int -7x^{6}\mathrm{d}x+\int x^{2}\mathrm{d}x$$

Iffattura ‘l barra l-kostanti f’kull wieħed minn dawn it-termini.

$$20\int x^{9}\mathrm{d}x-7\int x^{6}\mathrm{d}x+\int x^{2}\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{9}\mathrm{d}x ma' \frac{x^{10}}{10}. Immultiplika 20 b'\frac{x^{10}}{10}.

$$2x^{10}-7\int x^{6}\mathrm{d}x+\int x^{2}\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{6}\mathrm{d}x ma' \frac{x^{7}}{7}. Immultiplika -7 b'\frac{x^{7}}{7}.

$$2x^{10}-x^{7}+\int x^{2}\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{2}\mathrm{d}x ma' \frac{x^{3}}{3}.

$$2x^{10}-x^{7}+\frac{x^{3}}{3}$$

Jekk F\left(x\right) huwa antiderivattiv ta' f\left(x\right), allura s-sett tal-antiderivati kollha ta' f\left(x\right) jingħata minn F\left(x\right)+C. Għalhekk, żid il-kostanti ta' integrazzjoni C\in \mathrm{R} mar-riżultat.

$$2x^{10}-x^{7}+\frac{x^{3}}{3}+С$$

Iddifferenzja w.r.t. x

$$20x^{9}-7x^{6}+x^{2}$$